Optimal. Leaf size=267 \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)} \]
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Rubi [A] time = 0.32, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac {32 c^3 d^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{3003 (d+e x)^{7/2} (f+g x)^{7/2} (c d f-a e g)^4}+\frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{429 (d+e x)^{7/2} (f+g x)^{9/2} (c d f-a e g)^3}+\frac {12 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{143 (d+e x)^{7/2} (f+g x)^{11/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{13 (d+e x)^{7/2} (f+g x)^{13/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rule 872
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{15/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {(6 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{13/2}} \, dx}{13 (c d f-a e g)}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {\left (24 c^2 d^2\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{11/2}} \, dx}{143 (c d f-a e g)^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {\left (16 c^3 d^3\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{9/2}} \, dx}{429 (c d f-a e g)^3}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{13 (c d f-a e g) (d+e x)^{7/2} (f+g x)^{13/2}}+\frac {12 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{143 (c d f-a e g)^2 (d+e x)^{7/2} (f+g x)^{11/2}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{429 (c d f-a e g)^3 (d+e x)^{7/2} (f+g x)^{9/2}}+\frac {32 c^3 d^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{3003 (c d f-a e g)^4 (d+e x)^{7/2} (f+g x)^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 162, normalized size = 0.61 \[ \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (-231 a^3 e^3 g^3+63 a^2 c d e^2 g^2 (13 f+2 g x)-7 a c^2 d^2 e g \left (143 f^2+52 f g x+8 g^2 x^2\right )+c^3 d^3 \left (429 f^3+286 f^2 g x+104 f g^2 x^2+16 g^3 x^3\right )\right )}{3003 \sqrt {d+e x} (f+g x)^{13/2} (c d f-a e g)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.05, size = 1648, normalized size = 6.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 260, normalized size = 0.97 \[ -\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+56 a \,c^{2} d^{2} e \,g^{3} x^{2}-104 c^{3} d^{3} f \,g^{2} x^{2}-126 a^{2} c d \,e^{2} g^{3} x +364 a \,c^{2} d^{2} e f \,g^{2} x -286 c^{3} d^{3} f^{2} g x +231 a^{3} e^{3} g^{3}-819 a^{2} c d \,e^{2} f \,g^{2}+1001 a \,c^{2} d^{2} e \,f^{2} g -429 f^{3} c^{3} d^{3}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{3003 \left (g x +f \right )^{\frac {13}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (e x +d \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.12, size = 627, normalized size = 2.35 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {462\,a^6\,e^6\,g^3-1638\,a^5\,c\,d\,e^5\,f\,g^2+2002\,a^4\,c^2\,d^2\,e^4\,f^2\,g-858\,a^3\,c^3\,d^3\,e^3\,f^3}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {x^3\,\left (-10\,a^3\,c^3\,d^3\,e^3\,g^3+78\,a^2\,c^4\,d^4\,e^2\,f\,g^2-286\,a\,c^5\,d^5\,e\,f^2\,g+858\,c^6\,d^6\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^6\,d^6\,x^6}{3003\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c^4\,d^4\,x^4\,\left (3\,a^2\,e^2\,g^2-26\,a\,c\,d\,e\,f\,g+143\,c^2\,d^2\,f^2\right )}{3003\,g^5\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^5\,d^5\,x^5\,\left (a\,e\,g-13\,c\,d\,f\right )}{3003\,g^4\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a^2\,c\,d\,e^2\,x\,\left (567\,a^3\,e^3\,g^3-2093\,a^2\,c\,d\,e^2\,f\,g^2+2717\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {2\,a\,c^2\,d^2\,e\,x^2\,\left (371\,a^3\,e^3\,g^3-1469\,a^2\,c\,d\,e^2\,f\,g^2+2145\,a\,c^2\,d^2\,e\,f^2\,g-1287\,c^3\,d^3\,f^3\right )}{3003\,g^6\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^6\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^6}+\frac {6\,f\,x^5\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}+\frac {6\,f^5\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^5}+\frac {15\,f^2\,x^4\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {20\,f^3\,x^3\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^3}+\frac {15\,f^4\,x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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